# American Institute of Mathematical Sciences

May  2015, 20(3): 833-852. doi: 10.3934/dcdsb.2015.20.833

## Stochastic dynamics in a fluid--plate interaction model with the only longitudinal deformations of the plate

 1 Department of Mechanics and Mathematics, Karazin Kharkov National University, Kharkov, 61022, Ukraine 2 Institut für Mathematik, Institut für Stochastik, Ernst Abbe Platz 2, 07737, Jena, Germany

Received  September 2013 Revised  February 2014 Published  January 2015

We consider a stochastically perturbed coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a random pullback attractor of finite fractal dimension. Our argument is based on some modification of the method of quasi-stability estimates recently developed for deterministic systems.
Citation: Igor Chueshov, Björn Schmalfuß. Stochastic dynamics in a fluid--plate interaction model with the only longitudinal deformations of the plate. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 833-852. doi: 10.3934/dcdsb.2015.20.833
##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [2] G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. Optim., 55 (2007), 163-184. doi: 10.1007/s00245-006-0884-z. [3] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. [4] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in Fluids and waves, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007, 55-82. doi: 10.1090/conm/440/08476. [5] H. Bauer, Probability Theory, Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author, de Gruyter Studies in Mathematics, 23, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110814668. [6] P. Boxler, Stochastisch Zentrumsmannigfaltigkeiten, Thesis, Bremen, 1988. [7] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin, 1977. doi: 10.1007/BFb0087685. [8] A. Chambolle, B. Desjardins, M. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404. doi: 10.1007/s00021-004-0121-y. [9] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. [10] I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277. [11] I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Methods Appl. Sci., 34 (2011), 1801-1812. doi: 10.1002/mma.1496. [12] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x. [13] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. [14] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Well-posedness and long-time dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9. [15] I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynam. Differential Equations, 13 (2001), 355-380. doi: 10.1023/A:1016684108862. [16] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Commun. Pure Appl. Anal., 12 (2013), 1635-1656. doi: 10.3934/cpaa.2013.12.1635. [17] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Differential Equations, 254 (2013), 1833-1862. doi: 10.1016/j.jde.2012.11.006. [18] I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, in System Modeling and Optimization (25th IFIP TC7 Conference, Berlin, Sept.2011) (eds. D. Hömberg and F. Tröltzsch), IFIP Advances in Information and Communication Technology, 391, Springer, Berlin, 2013, 328-337. doi: 10.1007/978-3-642-36062-6_33. [19] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, Second edition, Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9. [20] G. P. Galdi, C. G. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$, Math. Ann., 331 (2005), 41-74. doi: 10.1007/s00208-004-0573-7. [21] M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech., 10 (2008), 388-401. doi: 10.1007/s00021-006-0236-4. [22] M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model, Appl. Anal., 88 (2009), 1053-1065. doi: 10.1080/00036810903114841. [23] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [24] N. D. Kopachevskii and Y. S. Pashkova, Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane, Russian J. Math. Phys., 5 (1997), 459-472 (1998). [25] J. E. Lagnese, Modelling and stabilization of nonlinear plates, in Estimation and Control of Distributed Parameter Systems (Vorau, 1990), Internat. Ser. Numer. Math., 100, Birkhäuser, Basel, 1991, 247-264. [26] J. E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 6, Masson, Paris, 1988. [27] J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl. (9), 85 (2006), 269-294. doi: 10.1016/j.matpur.2005.08.001. [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [29] T. J. Pedley, The Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge, 1980. doi: 10.1017/CBO9780511896996. [30] K. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1983. doi: 10.1017/CBO9780511608728. [31] B. Schmalfuß, The random attractor of the stochastic Lorenz system, Z. Angew. Math. Phys., 48 (1997), 951-975. doi: 10.1007/s000330050074. [32] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [33] H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, [2013 reprint of the 2001 original] [MR1928881], Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2001. [34] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. [35] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second edition, Johann Ambrosius Barth, Heidelberg, 1995. [36] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.

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##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [2] G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. Optim., 55 (2007), 163-184. doi: 10.1007/s00245-006-0884-z. [3] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. [4] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in Fluids and waves, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007, 55-82. doi: 10.1090/conm/440/08476. [5] H. Bauer, Probability Theory, Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author, de Gruyter Studies in Mathematics, 23, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110814668. [6] P. Boxler, Stochastisch Zentrumsmannigfaltigkeiten, Thesis, Bremen, 1988. [7] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin, 1977. doi: 10.1007/BFb0087685. [8] A. Chambolle, B. Desjardins, M. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404. doi: 10.1007/s00021-004-0121-y. [9] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. [10] I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277. [11] I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Methods Appl. Sci., 34 (2011), 1801-1812. doi: 10.1002/mma.1496. [12] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x. [13] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. [14] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Well-posedness and long-time dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9. [15] I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynam. Differential Equations, 13 (2001), 355-380. doi: 10.1023/A:1016684108862. [16] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Commun. Pure Appl. Anal., 12 (2013), 1635-1656. doi: 10.3934/cpaa.2013.12.1635. [17] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Differential Equations, 254 (2013), 1833-1862. doi: 10.1016/j.jde.2012.11.006. [18] I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, in System Modeling and Optimization (25th IFIP TC7 Conference, Berlin, Sept.2011) (eds. D. Hömberg and F. Tröltzsch), IFIP Advances in Information and Communication Technology, 391, Springer, Berlin, 2013, 328-337. doi: 10.1007/978-3-642-36062-6_33. [19] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, Second edition, Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9. [20] G. P. Galdi, C. G. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$, Math. Ann., 331 (2005), 41-74. doi: 10.1007/s00208-004-0573-7. [21] M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech., 10 (2008), 388-401. doi: 10.1007/s00021-006-0236-4. [22] M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model, Appl. Anal., 88 (2009), 1053-1065. doi: 10.1080/00036810903114841. [23] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [24] N. D. Kopachevskii and Y. S. Pashkova, Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane, Russian J. Math. Phys., 5 (1997), 459-472 (1998). [25] J. E. Lagnese, Modelling and stabilization of nonlinear plates, in Estimation and Control of Distributed Parameter Systems (Vorau, 1990), Internat. Ser. Numer. Math., 100, Birkhäuser, Basel, 1991, 247-264. [26] J. E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 6, Masson, Paris, 1988. [27] J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl. (9), 85 (2006), 269-294. doi: 10.1016/j.matpur.2005.08.001. [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [29] T. J. Pedley, The Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge, 1980. doi: 10.1017/CBO9780511896996. [30] K. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1983. doi: 10.1017/CBO9780511608728. [31] B. Schmalfuß, The random attractor of the stochastic Lorenz system, Z. Angew. Math. Phys., 48 (1997), 951-975. doi: 10.1007/s000330050074. [32] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [33] H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, [2013 reprint of the 2001 original] [MR1928881], Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2001. [34] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. [35] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second edition, Johann Ambrosius Barth, Heidelberg, 1995. [36] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.
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